2 deleted 22 characters in body
source | link

Perhaps:

Show[ContourPlot[R ω p Sin[ω Min[τ, 40 - τ]] + R ω p α - 9/4 r p R^3 ω - η p R == 0,
 {τ, #, #2}, {R, 0, 2}, ContourShading -> None, ContourStyle -> Directive[#3, Thick]]&@@@Thick]]& @@@
  {{0, 20, Blue}, {20, 40, Green}}, 
 FrameTicks->{{Automatic, Automatic}, {{#, Min[#, 40-#]}&/@Range[0, 40, 10], Automatic}}, 
 PlotRange -> {All, {0, 2}}]

enter image description here

Perhaps:

Show[ContourPlot[R ω p Sin[ω Min[τ, 40 - τ]] + R ω p α - 9/4 r p R^3 ω - η p R == 0,
 {τ, #, #2}, {R, 0, 2}, ContourShading -> None, ContourStyle -> Directive[#3, Thick]]&@@@
  {{0, 20, Blue}, {20, 40, Green}}, 
 FrameTicks->{{Automatic, Automatic}, {{#, Min[#, 40-#]}&/@Range[0, 40, 10], Automatic}}, 
 PlotRange -> {All, {0, 2}}]

enter image description here

Perhaps:

Show[ContourPlot[R ω p Sin[ω Min[τ, 40 - τ]] + R ω p α - 9/4 r p R^3 ω - η p R == 0,
 {τ, #, #2}, {R, 0, 2},  ContourStyle -> Directive[#3, Thick]]& @@@
  {{0, 20, Blue}, {20, 40, Green}}, 
 FrameTicks->{{Automatic, Automatic}, {{#, Min[#, 40-#]}&/@Range[0, 40, 10], Automatic}}, 
 PlotRange -> {All, {0, 2}}]

enter image description here

1
source | link

Perhaps:

Show[ContourPlot[R ω p Sin[ω Min[τ, 40 - τ]] + R ω p α - 9/4 r p R^3 ω - η p R == 0,
 {τ, #, #2}, {R, 0, 2}, ContourShading -> None, ContourStyle -> Directive[#3, Thick]]&@@@
  {{0, 20, Blue}, {20, 40, Green}}, 
 FrameTicks->{{Automatic, Automatic}, {{#, Min[#, 40-#]}&/@Range[0, 40, 10], Automatic}}, 
 PlotRange -> {All, {0, 2}}]

enter image description here